Question

Consider a hyperbola $xy=4$ and a line $2x+y=4.$ Let the given line intersect the $x-$axis at $R.$ If a line through $R$ intersects the hyperbola at $S$ and $T,$ then the minimum value of $RS\times RT=$

Answer 1

Given line intersects the $x-$axis at $R(2,0).$
Any point on the line which passes through $R$ and at a distance of $r$ units from $R$ is
$(2+r\cos \theta ,r\sin \theta )$

If this point lies on hyperbola $xy=4,$ then we have $(2+r\cos \theta )\left(r\sin \theta \right)=4$
$\Rightarrow r^2\sin \theta \cos \theta +2r\sin \theta -4=0$
It is quadratic in terms of $r,$ whose roots are $RS,RT.$
Product of roots of the above quadratic equation in ${}^{\prime }{r}^{\prime }$ is $r_1{r}_2=RS\times RT=\frac{8}{|\sin 2\theta |}$, whose minimum value $=8(\because 0\le |\sin 2\theta |\le 1)$
$\therefore$ Minimum value of $RS\times RT=8$